Introduction

When I first learned about decision feedback equalization, one of the bullet points is, “it is a nonlinear equalizer”, but I never knew why. Today, I will answer the question:

What makes DFE a nonlinear equalizer?

Decision Feedback Equalization Technique

Shown in Fig. 1, decision feedback equalizer (DFE) can open a closed eye. Nonetheless, the signature of an opened DFE eye is different than other equalizations. There are kinks in the eye diagram. To examine the eye diagram a little closer, we apply single pulse analysis to look at the blink of an eye.

Just like the eye diagram, we expect the single pulse response after decision feedback equalization to also have kinks. Sure enough, in Fig. 2, we see the kinks in the equalized single pulse response.

Fig. 2: Equalized single pulse response shows how DFE corrects post-cursor ISI on a single pulse that has all 0’s but a single 1. DFE inserts negative amplitudes after the received “1” pulse to better detect the next 0.

Taking a closer look, one observes that since the single pulse has all 0’s but a single 1 in Fig. 2, as soon as DFE algorithm sees a 1, it tries to reduce inter-symbol interference (ISI) by adding negative amplitudes so that the following low voltage is lower, allowing better detection of the next 0.

By the same token, when we send a single pulse that has all 1’s but a single 0, we should expect that as soon as the algorithm sees a 0, it tries to reduce ISI by adding positive amplitudes, allowing better detection of the next 1.

Fig. 3: Equalized single pulse response shows how DFE corrects post-cursor ISI on a single pulse that has all 1’s but a single 0. DFE inserts positive amplitudes after the received “0” pulse to better detect the next 1.

Result shown in Fig. 3 is consistent with our expectation. DFE algorithm is reducing ISI based on the detected data (symbol).

Fig. 4: Comparison between received waveform and equalized waveform shows how DFE acts on the received waveform.

By comparing the received waveform and waveform after DFE, as seen in Fig. 4, we can further see the action of DFE algorithm, but the question remains:

What makes DFE a nonlinear equalizer?

Symbol Detection and Decision: A Nonlinear Filter

At the arrival of received data (symbols), DFE algorithm detects and makes a decision. Assuming the decision is correct, proper tap values are chosen and feedback to the originally received data.

Fig. 5: An example of a symbol detector. Because the output does not scale linearly with the input, a symbol detector is nonlinear.

Shown in Fig. 5 is a symbol detection processing block. As the input doubles from 0.6 V to 1.2 V, the output does not double. Consequently, symbol detection is nonlinear. In turn, decision feedback equalization is also nonlinear.

But how do we make sure the detection is correct?

Fig. 6 is an illustration of DFE block diagram. The received symbol first undergo feedforward equalization so that the symbol detector can make the correct decisions. After the symbol detector makes a decision, the result goes through a feedback filter to be combined with the previously detected symbol.

Fig. 6: Decision Feedback Equalizer (DFE) block diagram. A feedforward filter is at the front end of DFE to help the symbol detector make a correct decision. Each decision then goes through feedback filter to be combined with previous symbol.

Because the input to the feedback filter consists of the sequence of decisions from previously detected symbols, which it uses to remove the portion of the ISI caused by those symbols, DFE only removes post-cursor ISI. Moreover, since DFE assumes that past symbol decisions are correct. Incorrect decisions from the symbol detector corrupt the filtering of the feedback loop. As a result, the inclusion of the feedforward filter on the front end is crucial in minimizing the probability of error [1].

Realization of Decision Feedback Equalization

Given the basic algorithm of DFE, I decided to design my own DFE, see Fig. 7.

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Fig. 7: Demonstration of a homemade DFE in Keysight ADS. See attached workspace for detail.

Knowing the input sequence is going to be a single “1” pulse, i.e. all 0's but and single 1, I first changed the feedback filter coefficients, V_tap1 and V_tap2, until the post-cursor ISI is reduced enough. Then, I adjusted the delay of the two taps so the corrections take place at the right time. When all was said and done, I had created a homemade 2-tap DFE. Fig. 8 shows the equalized single pulse response.

Fig. 8: Applying homemade 2-tap DFE to a single pulse.

In the process of creating a homemade DFE, I learned that DFE algorithm is not trivial. It requires many moving pieces to align. Besides the correct symbol detection, the timing and feedback filter coefficients (tap values) also need to be appropriately selected for different channels.

Fig. 9: DFE algorithm is readily available in Keysight ADS channel simulation. There are several adaptive algorithms to choose from.

Good news! To help expedite the simulating and testing process, DFE algorithms are implemented and readily available in ADS. ADS helps you test the amount of stress your channel can handle with DFE and adaptive DFE enabled, see Fig. 9.

Each of the equalizations has its own personality. While CTLE is sitting in the analog world, operating in the frequency domain, in the digital realm, FFE and DFE are working comfortably in the time domain.

Of course, each personality has its strength and weakness, and so does each equalization. In the near future, I will examine the pros and cons of equalization techniques. Make sure you bookmark the blog and check back regularly.